Low-rank factorization is used in many areas of computer science where one performs spectral analysis on large sensitive data stored in the form of matrices. In this paper, we study differentially private low-rank factorization of a matrix with respect to the spectral norm in the turnstile update model. In this problem, given an input matrix A ∈ Rm×n updated in the turnstile manner and a target...

ABSTRACT. In contemporary applied and computationalmathematics, a frequent challenge is to bound the expectation of the spectral norm of a sum of independent randommatrices. This quantity is controlled by the norm of the expected square of the randommatrix and the expectation of the maximum squared norm achieved by one of the summands; there is also aweak dependence on the dimension of the rand...

In this paper, the spectral expression of the frequency-limited H2-norm, firstly presented in [18], is extended to the case of LTI dynamical systems which are not diagonalizable. This extension is achieved by considering partial fraction decomposition of the transfer function associated to a system which is in a Jordan form. Besides, examples are presented to illustrate the behaviour of the fre...

In this paper the localization properties of the spectral expansions of distributions related to the self adjoint extension of the Schrodinger operator are investigated. Spectral decompositions of the distributions and some classes of distributions are defined. Estimations for Riesz means of the spectral decompositions of the distributions in the norm of the Sobolev classes with negative order ...

The theory of a class of spectral methods is extended to Volterra integrodifferential equations which contain a weakly singular kernel (t − s)−μ with 0 < μ < 1. In this work, we consider the case when the underlying solutions of weakly singular Volterra integro-differential equations are sufficiently smooth. We provide a rigorous error analysis for the spectral methods, which shows that both th...

Let Ω⊂ Rd be a bounded or an unbounded Lipschitz domain. In this note we address the problem of continuation of functions from the Sobolev space H1(Ω) up to functions in the Sobolev space H1(Rd) via a linear operator. The minimal possible norm of such an operator is estimated from below in terms of spectral properties of self-adjoint Robin Laplacians on domains Ω and Rd \Ω . Another estimate of...